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surfaces required the Calculus of Partial Differentials: the
isoperimetrieal problems resulted in the formation of the
Calculus of Variations. And reciprocally, all these great steps
in Algebraic Science had immediately their applications to
Geometry, and led to the discovery of new relations between
points or lines or surfaces. But even if the applications of the
method had not been so manifold and important, there would
still have been derivable a high intellectual pleasure from the
contemplation of it as a method.
'The first important application of this algebraical method of
co-ordinates to the study of optical systems was made by
Malus, a French officer of engineers in Napoleon's army in
Egypt, and who has acquired celebrity in the history of Phy-
sical Optics as the discoverer of polarization of light by reflec-
tion. Malus presented to the Institute of France, in 1807, a
profound mathematical work which is of the kind above alluded
to, and is entitled Traite cTOptique. The method employed in
that treatise may be thus described:- The  direction  of a
straight ray of any final optical system being considered as
dependent on the position of some assigned point on the ray,
according to some law which characterizes the particular
system and distinguishes it from others; this law may be
algebraically expressed by assigning three expressions for the
three co-ordinates of some other point of the ray, as functions
of the three co-ordinates of the point proposed. Malus accord-
ingly introduces general symbols denoting three such functions
for at least three functions equivalent to these), and proceeds
to draw several important general conclusions, by very compli-
cated yet symmetric calculations; many of which conclusions,
along with many others, were also obtained afterwards by
myself, when, by a method nearly similar, without knowing
what Malus had done, I began my own attempt to apply
Algebra to Optics. But my researches soon conducted me to
substitute, for this method of Malus, a very different, and (as I
conceive that I have proved) a much more appropriate one, for
the study of optical systems; by which, instead of employing
the three functions above mentioned, or at least their two ratios,
it becomes sufficient to employ one function, which I call